A363452 Total number of blocks containing only odd elements in all partitions of [n].

0, 1, 1, 5, 12, 62, 206, 1189, 4949, 31775, 156972, 1110280, 6301550, 48637701, 310279615, 2591820857, 18293310174, 164218811718, 1267153412532, 12152174863961, 101557600812015, 1035203191874931, 9299499328238110, 100314319611860936, 962663031508255416

Offset: 0

Author

Alois P. Heinz, Jun 02 2023

Keywords

• nonn

Examples

			a(3) = 5 = 0 + 1 + 1 + 1 + 2 : 123, 12|3, 13|2, 1|23, 1|2|3.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..250
• Wikipedia, Partition of a set

Crossrefs

Cf. A000110, A094577, A124420, A363434, A363453.

Programs

Maple

b:= proc(n, k) local g, u; g:= floor(n/2); u:=ceil(n/2);
      add(Stirling2(i, k)*binomial(u, i)*
      add(Stirling2(g, j)*j^(u-i), j=0..g), i=k..u)
    end:
a:= n-> add(b(n, k)*k, k=0..ceil(n/2)):
seq(a(n), n=0..25);
# second Maple program:
b:= proc(n, x, y, m) option remember; `if`(n=0, y,
      `if`(x+m>0, b(n-1, y, x, m)*(x+m), 0)+b(n-1, y, x+1, m)+
      `if`(y>0, b(n-1, y-1, x, m+1)*y, 0))
    end:
a:= n-> b(n, 0$3):
seq(a(n), n=0..25);

Mathematica

b[n_, x_, y_, m_] := b[n, x, y, m] = If[n == 0, y,
     If[x + m > 0, b[n-1, y, x, m]*(x+m), 0] + b[n-1, y, x+1, m] +
     If[y > 0, b[n-1, y-1, x, m+1]*y, 0]];
a[n_] := b[n, 0, 0, 0];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Dec 08 2023, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=0..ceiling(n/2)} k * A124420(n,k).

a(n) = A363434(n) - A363453(n).

a(2n) = A363453(2n).

a(2n+1) = A363453(2n+1) + A094577(n).